{"paper":{"title":"On Nathanson's Triangular Number Phenomenon","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Kevin O'Bryant","submitted_at":"2025-06-25T21:06:16Z","abstract_excerpt":"For a finite set $A\\subseteq \\mathbb{Z}$, the $h$-fold sumset is $hA :=\\{x_1+\\dots+x_h:x_i\\in A\\}$. We interpret the beginning of the sequence of sumset sizes $(|hA|)_{h=1}^\\infty$ in terms of the successive $L^1$-minima of a lattice (specifically, the points in $\\mathbb{Z}^{|A|}$ whose coordinates sum to 0 and which are perpendicular to $\\langle a_1,\\dots,a_{|A|}\\rangle$). In particular, if $h_1,h_2$ are the first and second minima, and $1\\le h<h_1$, then $|hA|=\\binom{h+|A|-1}{|A|-1}$, while if $h_1\\le h <h_2$, then $|hA|=\\binom{h+|A|-1}{|A|-1}-\\binom{h-h_1+|A|-1}{|A|-1}$. This explains the a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.20836","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.20836/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}